scholarly journals Convergence Rates in the Implicit Renewal Theorem on Trees

2013 ◽  
Vol 50 (4) ◽  
pp. 1077-1088
Author(s):  
Predrag R. Jelenković ◽  
Mariana Olvera-Cravioto
Keyword(s):  
Fixed Point ◽  
Rate Of Convergence ◽  
Convergence Rates ◽  
Tail Asymptotics ◽  
Renewal Theorem ◽  
Branching Trees

We consider possibly nonlinear distributional fixed-point equations on weighted branching trees, which include the well-known linear branching recursion. In Jelenković and Olvera-Cravioto (2012), an implicit renewal theorem was developed that enables the characterization of the power-tail asymptotics of the solutions to many equations that fall into this category. In this paper we complement the analysis in our 2012 paper to provide the corresponding rate of convergence.

scholarly journals Convergence Rates in the Implicit Renewal Theorem on Trees

2013 ◽  
Vol 50 (04) ◽  
pp. 1077-1088
Author(s):  
Predrag R. Jelenković ◽  
Mariana Olvera-Cravioto
Keyword(s):  
Fixed Point ◽  
Rate Of Convergence ◽  
Convergence Rates ◽  
Tail Asymptotics ◽  
Renewal Theorem ◽  
Branching Trees

We consider possibly nonlinear distributional fixed-point equations on weighted branching trees, which include the well-known linear branching recursion. In Jelenković and Olvera-Cravioto (2012), an implicit renewal theorem was developed that enables the characterization of the power-tail asymptotics of the solutions to many equations that fall into this category. In this paper we complement the analysis in our 2012 paper to provide the corresponding rate of convergence.

scholarly journals Implicit Renewal Theory and Power Tails on Trees

2012 ◽  
Vol 44 (2) ◽  
pp. 528-561 ◽  
Author(s):  
Predrag R. Jelenković ◽  
Mariana Olvera-Cravioto
Keyword(s):  
Random Vector ◽  
Renewal Theory ◽  
Tail Asymptotics ◽  
Renewal Theorem ◽  
Branching Trees ◽  
Independent And Identically Distributed

We extend Goldie's (1991) implicit renewal theorem to enable the analysis of recursions on weighted branching trees. We illustrate the developed method by deriving the power-tail asymptotics of the distributions of the solutions R to and similar recursions, where (Q, N, C1, C2,…) is a nonnegative random vector with N ∈ {0, 1, 2, 3,…} ∪ {∞}, and are independent and identically distributed copies of R, independent of (Q, N, C1, C2,…); here ‘∨’ denotes the maximum operator.

scholarly journals Implicit Renewal Theory and Power Tails on Trees

2012 ◽  
Vol 44 (02) ◽  
pp. 528-561 ◽  
Author(s):  
Predrag R. Jelenković ◽  
Mariana Olvera-Cravioto
Keyword(s):  
Random Vector ◽  
Renewal Theory ◽  
Tail Asymptotics ◽  
Renewal Theorem ◽  
Branching Trees ◽  
Independent And Identically Distributed

We extend Goldie's (1991) implicit renewal theorem to enable the analysis of recursions on weighted branching trees. We illustrate the developed method by deriving the power-tail asymptotics of the distributions of the solutionsRtoand similar recursions, where (Q,N,C1,C2,…) is a nonnegative random vector withN∈ {0, 1, 2, 3,…} ∪ {∞}, andare independent and identically distributed copies ofR, independent of (Q,N,C1,C2,…); here ‘∨’ denotes the maximum operator.

scholarly journals Fixed Points of the Evacuation of Maximal Chains on Fuss Shapes

10.37236/6898 ◽  
2018 ◽  
Vol 25 (1) ◽  
Author(s):  
Sen-Peng Eu ◽  
Tung-Shan Fu ◽  
Hsiang-Chun Hsu ◽  
Yu-Pei Huang
Keyword(s):  
Fixed Point ◽  
Fixed Points ◽  
Hasse Diagram ◽  
Linear Extensions ◽  
Rectangular Shape ◽  
Explicit Characterization ◽  
Finite Poset ◽  
Maximal Chains

For a partition $\lambda$ of an integer, we associate $\lambda$ with a slender poset $P$ the Hasse diagram of which resembles the Ferrers diagram of $\lambda$. Let $X$ be the set of maximal chains of $P$. We consider Stanley's involution $\epsilon:X\rightarrow X$, which is extended from Schützenberger's evacuation on linear extensions of a finite poset. We present an explicit characterization of the fixed points of the map $\epsilon:X\rightarrow X$ when $\lambda$ is a stretched staircase or a rectangular shape. Unexpectedly, the fixed points have a nice structure, i.e., a fixed point can be decomposed in half into two chains such that the first half and the second half are the evacuation of each other. As a consequence, we prove anew Stembridge's $q=-1$ phenomenon for the maximal chains of $P$ under the involution $\epsilon$ for the restricted shapes.

A Characterization of the Compound-Exponential Type Distributions

2011 ◽  
Vol 54 (3) ◽  
pp. 464-471
Author(s):  
Tea-Yuan Hwang ◽  
Chin-Yuan Hu
Keyword(s):  
Fixed Point ◽  
Exponential Type ◽  
Existence And Uniqueness ◽  
Fixed Point Equation ◽  
Point Equation

AbstractIn this paper, a fixed point equation of the compound-exponential type distributions is derived, and under some regular conditions, both the existence and uniqueness of this fixed point equation are investigated. A question posed by Pitman and Yor can be partially answered by using our approach.

A Characterization of the Hughes Planes

1965 ◽  
Vol 17 ◽  
pp. 916-922 ◽  
Author(s):  
T. G. Ostrom
Keyword(s):  
Fixed Point ◽  
Projective Plane ◽  
Related Notion ◽  
Fixed Line

Baer (1) introduced the term "(p,L)-collineation" to denote a central collineation with centre p and axis L. We shall find it convenient to use a modification of the related notion of "(p, L)-transitivity."Definition. Let π0 be a subplane of the projective plane π. Let L be a fixed line of π0, and let p be a fixed point of π0. Let r and s be any two points of π0 that are collinear with p, distinct from p, and not on L. If, for each such choice of r and s, there is a (p, L)-collineation of π that (1) carries π0 into itself and (2) carries r into s, we shall say that π is (p, L, π0)-transitive.

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